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Nonlinear Hodge correspondence in positive characteristic

Mao Sheng

TL;DR

This work constructs a nonlinear analogue of the nonabelian Hodge correspondence in positive characteristic by equating transversal foliated S-varieties (nonlinear flat bundles) with nonlinear Higgs bundles, extending Cartier descent and Ogus–Vologodsky-type correspondences to the nonlinear setting. It develops both local and global nonlinear Hodge theories using $W_2(k)$-liftings, Frobenius liftings, and exponential twisting, and introduces nonlinear Fontaine modules as a natural bridge between the nonlinear flat and Higgs sides. The framework generalizes the linear OV correspondence to nonlinear objects and provides a robust platform for nonlinear nonabelian Hodge theory in characteristic $p$, including a globalization strategy for $G$-equivariant data. An appendix further extends nonlinear Cartier descent to arbitrary morphisms and offers a new proof of Ekedahl's correspondence via nonlinear Cartier descent, tying inseparable morphisms to $p$-closed foliations in a unified manner.

Abstract

In this article, we extend the nonabelian Hodge correspondence in positive characteristic to the nonlinear setting.

Nonlinear Hodge correspondence in positive characteristic

TL;DR

This work constructs a nonlinear analogue of the nonabelian Hodge correspondence in positive characteristic by equating transversal foliated S-varieties (nonlinear flat bundles) with nonlinear Higgs bundles, extending Cartier descent and Ogus–Vologodsky-type correspondences to the nonlinear setting. It develops both local and global nonlinear Hodge theories using -liftings, Frobenius liftings, and exponential twisting, and introduces nonlinear Fontaine modules as a natural bridge between the nonlinear flat and Higgs sides. The framework generalizes the linear OV correspondence to nonlinear objects and provides a robust platform for nonlinear nonabelian Hodge theory in characteristic , including a globalization strategy for -equivariant data. An appendix further extends nonlinear Cartier descent to arbitrary morphisms and offers a new proof of Ekedahl's correspondence via nonlinear Cartier descent, tying inseparable morphisms to -closed foliations in a unified manner.

Abstract

In this article, we extend the nonabelian Hodge correspondence in positive characteristic to the nonlinear setting.

Paper Structure

This paper contains 9 sections, 27 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Linear $\lambda$-connections on linear schemes
  3. Nonlinear Cartier descent theorem
  4. Local nonlinear Hodge correspondence
  5. Global nonlinear Hodge correspondence
  6. Nonlinear Fontaine modules
  7. Appendix: Nonlinear Cartier descent and Ekedahl's correspondence, by Mao Sheng and Jinxing Xu
  8. Nonlinear Cartier descent
  9. Ekedahl's correspondence via nonlinear Cartier descent

Key Result

Theorem 1.4

Suppose $S$ is smooth over $k$. Then there is an explicit equivalence of categories between the category $\mathrm{NMIC}_0(S)$ of transversal foliated $S$-varieties with vanishing $p$-curvature and the category $\mathrm{NHIG}_0(S')$ of Higgs $S'$-varieties with zero Higgs field, which satisfies the f

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Theorem 1.4: Theorem \ref{['equivalence in the case of vanishing curvature']}, Theorem \ref{['thm:nonlinear Cartier descent in appendix']}
  • Theorem 1.5: Theorem \ref{['local nonlinear Hodge correspondence']}
  • Theorem 1.6: Theorem \ref{['global nonlinear Hodge correspondence']}
  • Remark 1.7
  • Theorem 1.8: Theorem \ref{['global one-periodicity']}
  • Proposition 2.1
  • proof
  • ...and 55 more